- #1

Telemachus

- 835

- 30

a) Given the equation

[tex]\frac{d^2y}{dt^2}+\frac{1}{t}\frac{dy}{dt}+4t^2y(t)=0[/tex]

Use the substitution [tex]x=t^2[/tex] to find the general solution

b) Find the particular solution that verifies [tex]y(0)=5[/tex]

c) Does any solution accomplish [tex]y'(0)=2[/tex]? Justify.

Well, so what I did is:

[tex]x=t^2 \rightarrow t=\sqrt{x}[/tex]

[tex]\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}=\frac{dy}{dx}2t[/tex]

[tex]\frac{d^2y}{dt^2}=\frac{d^2y}{dx^2}4t^2+2\frac{dy}{dx}=4x\frac{d^2y}{dx^2}+2\frac{dy}{dx}[/tex]

Then [tex]\frac{d^2y}{dt^2}+\frac{1}{t}\frac{dy}{dt}+4t^2y(t)=\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}+y(x)=0[/tex]

Now I'm not pretty sure what I should do to solve this. I thought of using Frobenius, would that be right?